without loss of generality [1]},
where \({\mbox{$u$}_{\rm c}}= \mbox{$\Omega $}_* \times \mbox{$r$}\)  is the corotation vector,
\({\mbox{$B$}}\)  is the magnetic field,
and
\(\Phi \)  is a scalar potential called the “non-corotational electric potential”.
The first term of the equation is the corotational electric field
\({\mbox{$E$}}_{\rm c} = - {\mbox{$u$}_{\rm c}}\times {\mbox{$B$}}/c\) .
The field-aligned electric field is given by
\(E_\parallel = - \hat{{\mbox{$B$}}} \cdot \nabla \Phi \) ,
where
\(\hat{{\mbox{$B$}}}={\mbox{$B$}}/ |{\mbox{$B$}}|\)  is the unit vector along the magnetic field.
Some useful formulae under the steadiness condition are given in the appendix.
